Geometric projection of stochastic differentialequations

John Armstrong (King’s College London)Damiano Brigo (Imperial)

November 2, 2018

Idea: Projection

Idea: Projection

I Projection gives a method of systematically reducing thedimension of an ODE

I Projection onto a linear subspace is the standard numericalmethod for solving PDEs

I Projecting onto a curved manifold may be more effective if weknow the solution is close to this manifold

I e.g. perhaps the known soliton solutions to the KdV equationmight give good approximations to solutions to a pertubedKdeV equation?

This talk

I Question: How should the notion of projection be extended tostochastic differential equations?

I Answer:I There is a Stratonovich Projection which is best understood

using Stratonovich calculus.I There is an Extrinsic Ito Projection which is best understood

using Ito calculus.I There is an Intrinsic Ito Projection which is best understood by

using jet bundles.I . . .

I We willI Define these various notions of projection and discuss their

motivation and theoretical justificationsI Describe a geometric formulation of SDEs using 2-jets to

understand the Intrinsic Ito projectionI Look at some numerical results when projection is applied to

nonlinear filtering

Setup

I M is a submanifold of Rr

I ψ : U → Rn is a chart for MI φ = ψ−1

I We have an SDE on Rr

dXt = a dt +∑α

bα dWαt , X0

and want to approximate this using an SDE on Rn.

Definition: Stratonovich projection

1. Write the SDE in Stratonovich form

dXt = a(Xt)dt +∑α

bα(Xt) ◦ dW αt , X0

2. Apply the projection operator Π to each coefficient to obtainan SDE on M

dXt = ΠXta(Xt) dt +∑α

ΠXtbα(Xt) ◦ dW αt , ψ(X0)

Justifications

What are the justifications for using the Stratonovich projection?

I It is clearly a well defined SDE. (Contrast with projecting Itocoefficients)

I It is clearly generalizes projection of ODEs - i.e. when b = 0we get ODE projection.

I It gives good numerical results when applied to the filteringproblem

I It generalizes the Galerkin method which can be interpreted asprojection onto a linear subspace.

A justification for ODE projection

I Consider an ODE on Rr

dX

dt= a(X ), X0

I Look for an ODE on Rn of the form

dx

dt= a(x), ψ(X0)

such that|φ(xt)− Xt |2

is as small as possible.

A justification for ODE projection

I Compute Taylor expansion to see that leading term isminimized when:

a(ψ(x0)) = ψ∗ΠX0A(x0)

I Therefore ODE projection is the unique asymptotically optimalODE approximating the original ODE at all points on M.

I (Linear projection operator gives solution to a quadraticoptimization problem)

Repeat idea for SDEs

Equation in larger space Rr : Equation in chart:dX = a(X , t) dt + b(X , t)dWt dx = A(x , t)dt + B(x , t) dWt

We have Ito Taylor series estimates (Kloeden and Platen):

E (|Xt − φ(xt)|) = |b0 − φ∗B0|√t + O(t)

|E (Xt − φ(xt))| =

∣∣∣∣a0 − φ∗A0 −1

2(∇Bα,0φ∗)Bβ,0[W α,W β]

∣∣∣∣ t+ O(t2)

Extrinsic Ito Projection

To minimize first estimate:

φ∗B = Πb

If we define B like this for whole chart, second estimate isminimized when:

φ∗A = Πa− 1

2Π(∇Bαφ∗)Bβ[W α,W β]

I Given φ, define A and B using these equations

I This defines an SDE on the manifold

I We call this the Extrinsic Ito projection

I It is different from the Stratonovich projection

Discussion

I The Extinsic Ito Projection is optimal in the sense that itasymptotically minimizes two measures of the divergence ofthe approximation to the SDE from the true solution.

I Measure one is on the expectation of the absolute value. Thisdetermines the martingale part of our equation

I Measure two is on the absolute value of the expectation. Thisdetermines the bounded variation part of our equation

I The Extrinsic Ito Projection is “greedy” in that it finds thebest approximation over short time horizons and hopes theywill do well over long time horizons.

I Numerical test on the filtering problem indicate that it slightlyoutperforms the Stratonovich projection in practice overmoderate time horizons.

I Over longer time horizons, it is random which performs better.

Geodesic projection map

Let π denote the smooth map defined on a tubular neighbourhoodof M that projects Rr onto M along geodesics.

An alternative justification for ODE projection

I Consider an ODE on Rr

dX

dt= a(X ), X0

I Look for an ODE on Rn of the formdx

dt= a(x), ψ(X0)

such that|φ(xt)− Xt |2

d(xt , ψ ◦ π(Xt))

is as small as possible. d is induced Riemannian distance.

Intrinsic Ito projection

Repeating the ideas used to derive the Extrinsic Ito projection:

DefinitionThe Intrinsic Ito projection is the best approximation to π(Xt) inthe sense that it asymptotically minimizes both:

E (d(xt , ψ ◦ π((Xt)))

d(E (xt),E (ψ ◦ π(Xt)))

Discussion

All three projections are distinct. Which is better?

Lemma(Factorizable SDEs) Suppose that S is an SDE for X on Rr suchthat π(X ) solves an SDE S ′ on M then the Stratonovich andintrinic Ito projections are both equal to S ′. However, the extrinsicprojection may be different.

Example

The SDE S on R2

dXt = σYt dWt

dYt = σXt dWt

In polar coordinates, solutions satisfy:

dθ = −1

2σ2 sin(4θ) dt + σ cos(2θ) dWt

Understanding the Intrinsic Ito projection

DefinitionThe Intrinsic Ito projection is the best approximation to π(Xt) inthe sense that it asymptotically minimizes both:

E (d(xt , ψ ◦ π((Xt)))

d(E (xt),E (ψ ◦ π(Xt)))

I For applications, one must calculate this in local coordinates,but the resulting expression is complex

I One can understand this projection more intuitively, andexpress the answer more elegantly, using the language of2-jets.

Euler Scheme

I All being well in the limit the Euler scheme

δXt = a(X ) δt + b(X ) δWt

converges to a solution of the SDE

dXt = a(X )dt + b(X ) dWt

I d, δ, + imply vector space structure

I This is highly coordinate dependent

Curved SchemeLet γx be a choice of curve at each point x of M. γx(0) = x .

-2 -1 0 1 2 3

-2

-1

0

1

2

3

Consider the scheme

Xt+δt = γXt (δWt) X0

Concrete example

γE(x1,x2)(s) = (x1, x2) + s(−x2, x1) + 3s2(x1, x2)

-2 -1 0 1 2 3

-2

-1

0

1

2

3

I First order term is rotational vector

I Second order term is axial vector

Simulation: Large time step

γE(x1,x2)(s) = (x1, x2) + s(−x2, x1) + 3s2(x1, x2)

Simulation: Smaller time step

γE(x1,x2)(s) = (x1, x2) + s(−x2, x1) + 3s2(x1, x2)

Simulation: Even smaller

γE(x1,x2)(s) = (x1, x2) + s(−x2, x1) + 3s2(x1, x2)

Simulation: Convergence

γE(x1,x2)(s) = (x1, x2) + s(−x2, x1) + 3s2(x1, x2)

Formal argumentWrite:

γx(s) = x + γ′x(0)s +1

2γ′′x (0)s2 + O(s3)

Then:

Xt+δt = γt(δWt)

= Xt + γ′Xt(0)δWt +

1

2γ′′Xt(0)(δWt)

2 + O((δWt)

3)

Rearranging:

δXt = Xt+δt − Xt = γ′Xt(0)δWt +

1

2γ′′Xt(0)(δWt)

2 + O((δWt)

3)

Taking the limit:

dXt = b(X )dWt + a(X )(dWt)2 + O

((dWt)

3)

= b(X )dWt + a(X )dt

whereb(X ) = γ′X (0)

a(X ) = γ′′X (0)/2

Comments

I The curved scheme depends only on the 2-jet of the curve

I SDEs driven by 1-d Brownian motion are determined by 2-jetsof curves

I The first derivative determines the volatility term

I The second derivative determines the drift term

ODEs correspond to 1-jets of curvesSDEs correspond to 2-jets of curves

I Rigorous proof of convergence of quadratic scheme can beproved using standard results on Euler scheme

dXt = a(X )dt + b(X )dWt

= a(X )(d(W 2

t )− 2Wtd(Wt))

+ b(X )dWt

≈ a(X )(δ(W 2

t )− 2Wtδ(Wt))

+ b(X )δWt

= a(X )((δWt)

2)

+ b(X )δWt

I For general curved schemes some analysis needed.

Ito’s lemma

Given a family of curves γx we will write:

Xt ^ j2 (γx(dWt))

if Xt is the limit of our scheme.If

Xt ^ j2 (γx(dWt))

and f : X → Y then:

f (X )t ^ j2 (f ◦ γx(dWt))

Ito’s lemma is simply composition of functions.

Usual formulation

Xt ^ j2 (γx(dWt))

Is equivalent to:

dXt = a(X )dt + b(X )dWt , a(X ) =1

2γ′′X (0), b(X ) = γ′X (0)

We calculate the first two derivatives of f ◦ γX :

(f ◦ γX )′(t) =n∑

i=1

∂f

∂xi(γX (t))

dγXdt

(f ◦ γX )′′(t) =n∑

j=1

n∑i=1

∂2f

∂xi∂xj(γX (t))

dγ iXdt

dγjXdt

+n∑

i=1

∂f

∂xi(γX (t))

d2γXdt2

So f (Xt) ^ j2 (f ◦ γx(dWt)) is equivalent to standard Ito’s formula

Example

γE(x1,x2)(s) = (x1, x2) + s(−x2, x1) + 3s2(x1, x2)

-2 -1 0 1 2 3

-2

-1

0

1

2

3

Clearly polar coordinates might be a good idea. So consider thetransformation φ : R2/{0} → [−π, π]× R by:

φ(exp(s) cos(θ), exp(s) sin(θ)) = (θ, s),

The process j2(φ ◦ γE ) plotted using image manipulation software

The process j2(φ ◦ γE ) plotted by applying Ito’s lemma

d(θ, s) =

(0,

7

2

)dt + (1, 0) dWt .

Drawing SDEs

The following diagram commutes:

SDE for X SDE for f (X )

Picture of SDE for X in Rn f (Picture of SDE for X)

Ito’s lemma

Draw Drawf

Intrinsic Ito Projection: 2-jet formulationIf original SDE is:

Xt ^ j2 (γx(dWt))

then intrinsic Ito projection is:

xt ^ j2 (π ◦ γx(dWt))

Local coordinate formulation

Calculate Taylor series for π to second order to compute:

dx = Adt + Bα dWαt , x0

where:B iα = (π∗)

iβb

βα

and:

Ai = (π∗)iαa

α+(−1

2

∂2φγ

∂xα∂xβ(π∗)

aγ(π∗)

αδ (π∗)

βε

+∂2φε

∂xα∂xβ(π∗)

βδ h

aα − ∂2φγ

∂xα∂xβ(π∗)

βε (π∗)

ηγ(π∗)

ζδhηζh

aα

)× bδκb

ει [W

κ,W ι]t .

Numerical example

I The linear filtering problem has solutions given by Gaussiandistributions

I Maybe approximately linear filtering problems can be wellapproximated by Gaussian distributions?

I Heuristic algorithms:I Extended Kalman FilterI Ito Assumed Density FilterI Stratonovich Assumed Density FilterI Stratonovich Projection Filter

I Algortihms based on optimization arguments:I Extrinsic Ito Projection FilterI Intrinsic Ito Projection Filter

Relative performance (Hellinger Residuals)

All projections performed w.r.t. the Hellinger metric.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1

Time

Intrinsic Ito ProjectionExtrinsic Ito Projection

Stratonovich ProjectionIto ADF

Extended Kalman Filter

Summary - projection methods

Extrinsic Ito Intrinsic Ito Stratonovich

Optimal? Yes YesFactorizable SDE Surprising Expected Expected

Aesthetics ElegantPractice Best short term Best medium term Acceptable

I Note that our notion of optimal is based on expectation ofsquared residuals

I Other “risk measures” could be used

Summary - 2 jets

I 2-jets allow you to draw pictures of SDEs

I They provide an intuitive and elegant reformulation of Ito’slemma

I They provide an alternative route to coordinate free stochasticdifferential geometry to operator opproaches